Digital Signal Processing Questions and Answers – General Considerations for Design of Digital Filters

This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “General Consideration for Design of Digital Filters”.

1. The ideal low pass filter cannot be realized in practice.
a) True
b) False
View Answer

Answer: a
Explanation: We know that the ideal low pass filter is non-causal. Hence, a ideal low pass filter cannot be realized in practice.

2. The following diagram represents the unit sample response of which of the following filters?
Figure represents unit sample response of an ideal low pass filter at omega=pi/4
a) Ideal high pass filter
b) Ideal low pass filter
c) Ideal high pass filter at ω=π/4
d) Ideal low pass filter at ω=π/4
View Answer

Answer: d
Explanation: At n=0, the equation for ideal low pass filter is given as h(n)=ω/π.
From the given figure, h(0)=0.25=>ω=π/4.
Thus the given figure represents the unit sample response of an ideal low pass filter at ω=π/4.

3. If h(n) has finite energy and h(n)=0 for n<0, then which of the following are true?
a) \(\int_{-π}^π|ln⁡ |H(ω)||dω \gt -\infty\)
b) \(\int_{-π}^π|ln⁡ |H(ω)||dω \lt \infty\)
c) \(\int_{-π}^π|ln⁡|H(ω)||dω = \infty\)
d) None of the mentioned
View Answer

Answer: b
Explanation: If h(n) has finite energy and h(n)=0 for n<0, then according to the Paley-Wiener theorem, we have
\(\int_{-π}^π|ln⁡ |H(ω)||dω \lt \infty\)
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4. If |H(ω)| is square integrable and if the integral \(\int_{-\pi}^\pi |ln⁡|H(ω)||dω\) is finite, then the filter with the frequency response H(ω)=|H(ω)|ejθ(ω) is?
a) Anti-causal
b) Constant
c) Causal
d) None of the mentioned
View Answer

Answer: c
Explanation: If |H(ω)| is square integrable and if the integral \(\int_{-\pi}^\pi |ln⁡|H(ω)||dω\) is finite, then we can associate with |H(ω)| and a phase response θ(ω), so that the resulting filter with the frequency response H(ω)=|H(ω)|ejθ(ω) is causal.

5. The magnitude function |H(ω)| can be zero at some frequencies, but it cannot be zero over any finite band of frequencies.
a) True
b) False
View Answer

Answer: a
Explanation: One important conclusion that we made from the Paley-Wiener theorem is that the magnitude function |H(ω)| can be zero at some frequencies, but it cannot be zero over any finite band of frequencies, since the integral then becomes infinite. Consequently, any ideal filter is non-causal.
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6. If h(n) is causal and h(n)=he(n)+ho(n),then what is the expression for h(n) in terms of only he(n)?
a) h(n)=2he(n)u(n)+he(0)δ(n), n ≥ 0
b) h(n)=2he(n)u(n)+he(0)δ(n), n ≥ 1
c) h(n)=2he(n)u(n)-he(0)δ(n), n ≥ 1
d) h(n)=2he(n)u(n)-he(0)δ(n), n ≥ 0
View Answer

Answer: d
Explanation: Given h(n) is causal and h(n)= he(n)+ho(n)
=>he(n)=1/2[h(n)+h(-n)] Now, if h(n) is causal, it is possible to recover h(n) from its even part he(n) for 0≤n≤∞ or from its odd component ho(n) for 1≤n≤∞.
=>h(n)= 2he(n)u(n)-he(0)δ(n), n ≥ 0.

7. If h(n) is causal and h(n)=he(n)+ho(n),then what is the expression for h(n) in terms of only ho(n)?
a) h(n)=2ho(n)u(n)+h(0)δ(n), n ≥ 0
b) h(n)=2ho(n)u(n)+h(0)δ(n), n ≥ 1
c) h(n)=2ho(n)u(n)-h(0)δ(n), n ≥ 1
d) h(n)=2ho(n)u(n)-h(0)δ(n), n ≥ 0
View Answer

Answer: b
Explanation: Given h(n) is causal and h(n)= he(n)+ho(n)
=>he(n)=1/2[h(n)+h(-n)] Now, if h(n) is causal, it is possible to recover h(n) from its even part he(n) for 0≤n≤∞ or from its odd component ho(n) for 1≤n≤∞.
=>h(n)= 2ho(n)u(n)+h(0)δ(n), n ≥ 1
since ho(n)=0 for n=0, we cannot recover h(0) from ho(n) and hence we must also know h(0).
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8. If h(n) is absolutely summable, i.e., BIBO stable, then the equation for the frequency response H(ω) is given as?
a) HI(ω)-j HR(ω)
b) HR(ω)-j HI(ω)
c) HR(ω)+j HI(ω)
d) HI(ω)+j HR(ω)
View Answer

Answer: c
Explanation: If h(n) is absolutely summable, i.e., BIBO stable, then the frequency response H(ω) exists and
H(ω)= HR(ω)+j HI(ω)
where HR(ω) and HI(ω) are the Fourier transforms of he(n) and ho(n) respectively.

9. HR(ω) and HI(ω) are interdependent and cannot be specified independently when the system is causal.
a) True
b) False
View Answer

Answer: a
Explanation: Since h(n) is completely specified by he(n), it follows that H(ω) is completely determined if we know HR(ω). Alternatively, H(ω) is completely determined from HI(ω) and h(0). In short, HR(ω) and HI(ω) are interdependent and cannot be specified independently when the system is causal.
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10. What is the Fourier transform of the unit step function U(ω)?
a) πδ(ω)-0.5-j0.5cot(ω/2)
b) πδ(ω)-0.5+j0.5cot(ω/2)
c) πδ(ω)+0.5+j0.5cot(ω/2)
d) πδ(ω)+0.5-j0.5cot(ω/2)
View Answer

Answer: d
Explanation: Since the unit step function is not absolutely summable, it has a Fourier transform which is given by the equation
U(ω)= πδ(ω)+0.5-j0.5cot(ω/2).

11. The HI(ω) is uniquely determined from HR(ω) through the integral relationship. This integral is called as Continuous Hilbert transform.
a) True
b) False
View Answer

Answer: b
Explanation: If the HI(ω) is uniquely determined from HR(ω) through the integral relationship. This integral is called as discrete Hilbert transform.

12. The magnitude |H(ω)| cannot be constant in any finite range of frequencies and the transition from pass-band to stop-band cannot be infinitely sharp.
a) True
b) False
View Answer

Answer: a
Explanation: Causality has very important implications in the design of frequency-selective filters. One among them is the magnitude |H(ω)| cannot be constant in any finite range of frequencies and the transition from pass-band to stop-band cannot be infinitely sharp. This is a consequence of Gibbs phenomenon, which results from the truncation of h(n) to achieve causality.

13. The frequency ωP is called as ______________
a) Pass band ripple
b) Stop band ripple
c) Pass band edge ripple
d) Stop band edge ripple
View Answer

Answer: c
Explanation: Pass band edge ripple is the frequency at which the pass band starts transiting to the stop band.

14. Which of the following represents the bandwidth of the filter?
a) ωP+ ωS
b) -ωP+ ωS
c) ωPS
d) None of the mentioned
View Answer

Answer: b
Explanation: If ωP and ωS represents the pass band edge ripple and stop band edge ripple, then the transition width -ωP+ ωS gives the bandwidth of the filter.

Sanfoundry Global Education & Learning Series – Digital Signal Processing.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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